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Absolute Value
Objectives To find the absolute value of rational numbers;
and
to solve problems using absolute value.
a
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Common
Core State
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Ongoing Learning & Practice
Key Concepts and Skills
Solving Volume Problems
• Identify the absolute value of rational
numbers. Math Journal 2, p. 217C
Students practice finding volumes of
rectangular prisms.
[Number and Numeration Goal 1]
• Add and subtract signed numbers. [Operations and Computation Goal 1]
• Describe a general pattern with a
number sentence. [Patterns, Functions, and Algebra Goal 1]
Ongoing Assessment:
Recognizing Student Achievement
Use Math Journal 2, page 217C. [Measurement and Reference Frames
Goal 2]
Math Boxes 6 4a
Key Activities
Students define absolute value using a
distance model and algebraically. They find
the absolute value of rational numbers and
use absolute value to solve problems.
Key Vocabulary
absolute value
Math Journal 2, p. 217D
Students practice and maintain skills
through Math Box problems.
Study Link 6 4a
Math Masters, p. 192A
Students practice and maintain skills
through Study Link activities.
Materials
Math Journal 2, pp. 217A and 217B
Study Link 64
transparency of Math Masters, p. 417
Class Number Line
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Absolute Value Riddles
Math Masters, p. 192B
Students solve riddles involving
absolute value.
EXTRA PRACTICE
Playing a Variation of Top-It with
Positive and Negative Numbers
Student Reference Book, pp. 337 and 338
per partnership: complete deck of number
cards (the Everything Math Deck, if available)
Students play a variation of Top-It with
Positive and Negative Numbers that uses
absolute value.
ELL SUPPORT
Illustrating Terms
chart paper markers
Students create a poster that illustrates how
to find the absolute value of a number.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 p. 74
Lesson 6 4a
552A
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
6.NS.5, 6.NS.7, 6.NS.7a, 6.NS.7c, 6.NS.7d, 6.NS.8, 6.G.2
Mental Math and Reflexes
Math Message
Students solve computation problems with
signed numbers. Suggestions:
The distance from 0 to 1 on any number line is 1.
Find the distance of each of the following numbers
from 0. Use the Class Number Line.
9 + (-3) 6
26 ∗ (-1) -26
48 - (-22) 70
60 ÷ (-12) -5
3
1
3 ÷ -_
1 1_
or _
-_
4
2
2
a. 4 4
d. -8 8
b. 27 27
e. 0 0
c. -16 16
Study Link 6 4 Follow-Up
2
25
5
1
_
_
2 ∗_
-1 _
4 - 2 12 or - 12
3
Briefly go over the answers.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Have students explain how they used the number line to help
them find the distance between each number and zero. Ask
students to describe why each distance is positive or 0, even
though some of the numbers are negative. Distance is a positive
value or 0; there is no such thing as a negative distance. So the
distance from 0 to any nonzero number is positive. The distance
from 0 to 0 is 0.
Tell students that the distance between a number and 0 on the
number line is the absolute value of that number. The notation
for the absolute value of a number is | |. For example, we write
the absolute value of 5 as |5|.
If necessary, have students find the absolute value of several other
numbers using the Class Number Line. Suggestions:
●
|47| 47
●
●
|-18| 18
●
|-10.5| 10.5
3 | 20_
3
|20_
4
4
▶ Absolute Value
(Math Journal 2, p. 217A)
Circulate and assist as students work in pairs to complete
Problems 1–14 on the journal page.
552B Unit 6
Number Systems and Algebra Concepts
PARTNER
ACTIVITY
When most partnerships are finished, bring the class together to
share the general patterns that they wrote for Problems 13 and
14. Explain to students that the general patterns they identified
are another way of defining absolute value.
|x| =
{
x, if x ≥ 0
OPP(x), if x < 0
NOTE The notation for the algebraic definition of absolute value looks
complicated, but it is simply a more formal way of writing the general
patterns students identified in Problems 13 and 14 on journal page 217A.
Write the bracketed form of the definition on the board for students to see,
but do not insist that students write it this way.
Work through several examples using the algebraic definition of
absolute value. Suggestions:
●
|5| = 5 because 5 ≥ 0
●
|-8| = OPP(-8) = 8 because -8 < 0
▶ Absolute Value as a Magnitude
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 217A)
Explain to students that absolute value can be used to indicate
the magnitude, or size, of numbers in certain real-world contexts.
Sometimes, it is useful to compare the absolute values of numbers
instead of the numbers themselves.
Present this situation to students: Rita has a debt of $15 and
Jamal has a debt of $22. Ask students to use negative numbers
to represent these debts. Rita: -15; Jamal: -22 Ask students to
write an inequality that compares the debts. -22 < -15 Explain
that this number sentence seems to indicate that Jamal’s debt is
less than Rita’s debt. However, in everyday life, we would say that
Jamal has the greater debt because the magnitude of his debt is
greater: |-22| > |-15|, or 22 > 15.
Student Page
Date
Pose similar situations to students and ask them to use absolute
value to justify their answers. Suggestions:
●
●
●
The submarine Charlie is 1,500 feet under water, and the
submarine April is 2,300 feet under water. Which is deeper?
April is deeper; |-1,500| < |-2,300|
Shaun is 9 seconds behind the leader in a bike race. Kayla is
4 seconds behind the leader. Who is further behind?
Shaun; |-9| > |-4|
Larry’s Pizza sales dropped by 3.7% and Barb’s Pizza sales
dropped by 5.9%. Whose sales dropped more? Barb’s Pizza;
|-3.7%| < |-5.9%|
Time
LESSON
Absolute Value
6 4a
Solve.
1.
3.
|16| =
64
16
2.
= |-64|
4.
0.72
3.09
7. |-3.09| =
9. 21.741 = |21.741|
5.
11.
|0.72| =
6.
8.
10.
|-0.003| =
_1
2
4_1
3
0.003
94
1|
= |_
2
1
= |- 4_
3|
0
8,906 = |8,906|
|0| =
What do you notice about the absolute value of positive numbers and zero?
The absolute value of a positive number or zero is
the same as the number itself.
12.
What do you notice about the absolute value of negative numbers?
The absolute value of a negative number is the
opposite of the number.
Use algebraic notation to write the general pattern for each group of 3 special cases.
13.
|4.78| = 4.78
|0| = 0
|19,722| = 19,722
14.
General pattern:
Have students complete Problems 15 and 16 on journal page 217A.
If x ≥ 0,
|x | = x
|-270| = 270
15
15 = _
|-_
16
16 |
|-12.125| = 12.125
General pattern:
.
If x < 0,
|x | = OPP(x).
Answer the following questions. Use an inequality involving absolute value to justify your answer.
15.
Jayson charged $172 on his credit card for materials to start his lawn-mowing business.
Rita charged $98 on her credit card for materials to start her tutoring business.
Who has the greater debt?
16.
Jayson; |-172| > |-98|
The number of home sales in Illinois dropped 24% in one year while the number
of home sales in Indiana dropped 20%.
Which state had the larger decrease in home sales?
Illinois; |-24%| > |-20%|
Math Journal 2, p. 217A
205_246_EMCS_S_G6_MJ2_U06_576442.indd 217A
3/4/11 10:21 AM
Lesson 6 4a
552C
Student Page
Date
▶ Absolute Value and Distance
Time
LESSON
6 4a
Taxi Rides
(Math Masters, p. 417; Math Journal 2, p. 217B)
Jerissa and her aunt have planned a trip to the city. A map of the city is shown below.
Each grid square represents one block. They will be traveling around the city by taxi.
The taxi meter calculates the distance the taxi has traveled from the starting point (s )
to the ending point (e ) using the formula d = |s - e |.
94
Ask students to imagine that they are riding in a taxi. The fare
is based on the number of blocks that the taxi travels. The taxi
company has programmed the meter to calculate the distance the
taxi travels by entering the starting location s, entering the ending
location e, and then calculating s - e.
Jerissa has planned to visit different locations in the order shown below. Find the distance that
they will travel on each part of their trip. Use a number sentence to justify each answer.
y
10
9
Restaurant
Museum
8
WHOLE-CLASS
ACTIVITY
7
6
5
4
3
2
1
10
9
8
7
6
5
Shopping center
4
3
2
Theater
1 0
1
1
2
3
4
5
6
7
8
9
10
x
Display a transparency of Math Masters, page 417. Plot and label
Point A at (6,2) and Point B at (2,2). Ask: If you got in the taxi at
Point A and traveled to Point B, how many blocks would you have
traveled? 4 blocks How would the taxi meter find this distance?
Subtract the x-coordinates to find s - e: 6 - 2 = 4.
Train station
2
3
4
5
1.
9 blocks
8| = |-9| = 9
8 blocks
Museum to the restaurant:
Number sentence: |3 - (-5)| = |8| = 8
9 blocks
Restaurant to the theater:
| - (-1)| = |9| = 9
Number sentence: 8
2 blocks
Theater to the shopping center:
|-5 - (-7)| = |2| = 2
Number sentence:
10 blocks
Shopping center to the train station:
|-7 - 3| = |-10| = 10
Number sentence:
Train station to the museum:
Number sentence:
2.
3.
4.
5.
|-1 -
Next, plot and label Point C at (9,2). Tell students to imagine that
they got in the taxi at Point B and traveled to Point C. Ask: What
will happen when the taxi meter calculates this distance? There
will be a negative answer; 2 - 9 = -7. Can distance be negative?
No, distance is always a positive value or 0. How could absolute
value help in this situation? Taking the absolute value of the
difference would always give the distance that the taxi traveled.
Since distance is always positive or 0, finding the absolute value of
the difference of two numbers in a distance problem makes sense.
What formula could the taxi company program into the meter to
find the distance traveled? Sample answer: d = |s - e|
Math Journal 2, p. 217B
205_246_EMCS_S_G6_MJ2_U06_576442.indd 217B
3/4/11 10:21 AM
NOTE The formula used here for calculating
distance does take into account that each
location is an ordered pair. In this activity,
each pair of ordered pairs will have the same
x-coordinates or the same y-coordinates. The
coordinates that are the same can be ignored
because the difference between them is 0, so
they don’t contribute to the distance.
Plot several other points on the coordinate grid and have students
use the formula to find the distance between the two points,
imagining that they are in a taxi. Have students write a number
sentence to show how they used the formula to find the distance.
Suggestions:
(-3,1) to (-3,-7) 8 blocks; |1 - (-7)| = |8| = 8
(-7,-4) to (-7,5) 9 blocks; |(-4) - 5| = |-9| = 9
Student Page
Date
6 4a
(5,-8) to (10,-8) 5 blocks; |5 - 10| = |-5| = 5
Time
LESSON
Volume Problems
Leroy’s Moving and Storage has three different storage units for rent.
The table below shows the dimensions of each unit.
Unit Type
1.
Length
12 _1 feet
B
8 _12 feet
C
9 feet
2
10 _12 feet
Width
Height
8 _1 feet
15 feet
13 _3 feet
8 _12 feet
8 _12 feet
Find the volume of each of the storage units.
Show your work. Use the formulas at the right
to help you.
a.
b.
c.
2.
A
1,115_58 ft3
Volume of Unit A =
1,083_34 ft3
Volume of Unit B =
1,051_78 ft3
Volume of Unit C =
4
Have students work with a partner to solve the problems on
journal page 217B.
221
2
2 Ongoing Learning & Practice
Volume of a Rectangular Prism
V = B ∗ h or V = l ∗ w ∗ h
V = volume
B = area of base
l = length of base
w = width of base
h = height
▶ Solving Volume Problems
Jacqueline needs to rent a storage unit. The items she plans to store are currently in a trailer
1
1
_
with dimensions of 14_
2 feet by 8 3 feet by 9 feet. Which storage unit should Jacqueline rent?
Show your work and explain your thinking.
Sample answer: The volume of the trailer is 1,087_2 ft3.
1
3.
(Math Journal 2, p. 217C)
If Jacqueline’s items fill the trailer, Unit A would be
the only unit large enough to store all her items. If the
trailer is not full, she might be able to fit her items in
Unit B or Unit C.
1
1
_
_
Students review and apply the formula for the volume of a right
rectangular prism in the context of moving and storage problems.
Jarrett has packed all his items into boxes. The boxes are 2 2 feet in length, 2 2 feet in width,
and 2 feet in height. Jarrett has rented Unit B. How many boxes will Jarrett be able to fit in
Unit B? Explain your thinking.
Jarrett can place 72 boxes in Unit B. Three boxes will
fit along the length of the unit, and 6 boxes will fit along
the width of the unit. 3 ∗ 6 = 18 total boxes will fit on
the floor of the unit. He can stack 4 layers of boxes in
the unit. 4 ∗ 18 = 72 boxes
Math Journal 2, p. 217C
205_246_EMCS_S_G6_MJ2_U06_576442.indd 217C
552D Unit 6
INDEPENDENT
ACTIVITY
3/4/11 10:21 AM
Number Systems and Algebra Concepts
Student Page
Date
Journal
Page 217C
Ongoing Assessment:
Recognizing Student Achievement
Time
LESSON
Math Boxes
6 4a
1.
Simplify.
a.
2.
2
_
3 of 123
7
_
b. 6 of 72
Use journal page 217C, Problem 1 to assess students’ ability to use a
formula to find the volume of a rectangular prism. Students are making adequate
progress if they are able to calculate the volume of each of the storage units in
Problem 1. Some students may apply formulas to help them solve Problem 3.
c.
3
1
_
_
4 of 2 2
d.
7
4
_
_
8 of 7
82
84
15
_
_7
8 , or 1 8
_1
4
_
a. 5
3
_
c. 4
4 =
÷_
5
9
_
d. 18
16
3
÷_
=
18
3
87–89
3.
Give a ballpark estimate for each quotient.
93
4.
Sample estimates:
▶ Math Boxes 6 4a
1_35
2_29
15
_
1 =
÷_
2
8
2
_
_
b. 9 ÷ 5 =
2
[Measurement and Reference Frames Goal 2]
INDEPENDENT
ACTIVITY
Divide. Simplify if possible.
a.
487.8 ÷ 3
b.
619.725 5
c.
1,824 ÷ 60
d.
(Math Journal 2, p. 217D)
160
125
30
Complete each sentence using an
algebraic expression.
a.
Raven is 3 years older than her brother.
If Raven’s brother is b years old, then
b.
If every box of crayons contains
8 crayons, then c boxes of crayons
Raven is
60
5,478
_
92
contain
b+3
8∗c
years old.
crayons.
261
Mixed Review Math Boxes in this lesson are paired with
Math Boxes in Lessons 6-5 and 6-7. The skills in Problems
5 and 6 preview Unit 7 content.
▶ Study Link 6 4a
5.
Write each fraction as a decimal.
32
_
a. 100
1
_
b. 4
=
=
0.32
=
0.94
8
_
d. 20
=
0.4
INDEPENDENT
ACTIVITY
(Math Masters, p. 192A)
Suppose you toss a fair coin 10 times.
What is the probability of getting
HEADS on:
_1
0.25
47
_
c. 50
240
6.
a.
the first toss?
b.
the third toss?
c.
the tenth toss?
2
_1
2
_1
2
59–60,
74
148 149
Math Journal 2, p. 217D
205_246_EMCS_S_G6_MJ2_U06_576442.indd 217D
3/4/11 10:21 AM
Home Connection Students practice finding the absolute
value of numbers and use absolute value to calculate the
distance between points on a coordinate grid.
Study Link Master
Name
Date
STUDY LINK
Time
Absolute Value
6 4a
Plot and label the following points on the number line. The first one has
been done for you.
F
C
–10
E
–5
B
94
D
A
0
5
10
1.
Point A: 5.75
2.
0
Point B: _
10
3.
Point C: –6.5
4.
84
Point D: _
9
5.
Point E: –2.75
6.
1
Point F: – 8_
2
9.
| –6.5 | =
12.
1| =
| –8_
2
Find the absolute value of each of the numbers you plotted.
7.
10.
| 5.75 | =
84
84 | = 9
|_
9
5.75
_, or 9_1
3
8.
11.
0|=
|_
10
0
| –2.75 | =
2.75
Point A: (– 3,– 6)
14.
Point B: (1,– 6)
15.
Point C: (– 3,8)
16.
Point D: (– 8,8)
8_12
y
Plot the following points on the coordinate grid.
13.
6.5
D
10
C
9
8
7
6
5
4
3
2
1
10
9
8
7
6
5
4
3
2
1 0
1
1
2
3
4
5
6
7
8
9
10
2
3
Use absolute value to find the distance between
the given points. Show how you solved each
problem. Sample strategies are given.
17.
A and B
4; |–3 – 1| = |–4| = 4
18.
A and C
14; |8 – (–6)| = |14| = 14
19.
C and D
5; |–8 – (–3)| = |–5| = 5
A
4
5
6
B
7
8
9
10
Math Masters, p. 192A
180-216_EMCS_B_G6_MM_U06_576981.indd 192A
3/3/11 2:15 PM
Lesson 6 4a
552E
Teaching Master
Name
Date
LESSON
3 Differentiation Options
Time
Absolute Value Riddles
6 4a
Use your knowledge of absolute value to answer the riddles.
I am thinking of a number.
1.
The absolute value of the number is 19.
The number is not equal to its absolute value.
What number am I thinking of?
ENRICHMENT
-19
▶ Absolute Value Riddles
I am thinking of a number.
2.
I do not have to take an opposite to find the absolute value of the number.
The absolute value of the number is 451.
What number am I thinking of?
INDEPENDENT
ACTIVITY
15–30 Min
(Math Masters, p. 192B)
451
I am thinking of a number.
3.
Students extend their understanding of absolute value by solving
riddles involving absolute value. These riddles require students to
“work backward,” starting with the absolute value and thinking
about what the original number might have been. If students need
help getting started, you might use a simple example to show that
two numbers can have the same absolute value. For example,
|3| = |-3| = 3.
The absolute value of the number is not positive.
0
What number am I thinking of?
The balance of Amber’s bank account at the beginning of the day was $150.
At the end of the day, the absolute value of the change to her account
balance was $70.
4.
Amber deposited money today.
What is Amber’s new account balance?
$220
The absolute value of the change in what Hal’s car is worth since he bought
it is $7,500.
5.
Hal has not had any work done to increase what the car is worth.
Hal’s car was worth $16,000 when he bought it.
What is Hal’s car currently worth?
$8,500
Write and solve your own absolute value riddle below.
6.
Answers vary.
EXTRA PRACTICE
▶ Playing a Variation of
Math Masters, p. 192B
180-216_EMCS_B_G6_MM_U06_576981.indd 192B
3/3/11 2:15 PM
Top-It with Positive and
Negative Numbers
PARTNER
ACTIVITY
5–15 Min
(Student Reference Book, pp. 337 and 338)
To provide extra practice with absolute value, have students play a
variation of Top-It with Positive and Negative Numbers. Students
follow the directions found on Student Reference Book, pages 337
or 338. Instead of finding the sums or differences, students find
the absolute value of the sums or differences. The player with the
largest absolute value takes all the cards.
Student Page
ELL SUPPORT
Games
Top-It Games with Positive and Negative Numbers
▶ Illustrating Terms
SMALL-GROUP
ACTIVITY
5–15 Min
Materials □ 1 complete deck of number cards
To provide language support for absolute value, have students
create a poster that explains how to find the absolute value of a
number. The poster should reference both the number line and
algebraic definitions.
□ 1 calculator (optional)
Players
2 to 4
Skill
Addition and subtraction of positive and negative numbers
Object of the game To collect the most cards.
Addition Top-It with Positive and Negative Numbers
Directions
The color of the number on each card tells you if a card is
a positive number or a negative number.
♦ Black cards (spades and clubs) are positive numbers.
♦ Red cards (hearts and diamonds) or blue cards
(Everything Math Deck) are negative numbers.
1. Shuffle the deck and place it number-side down.
FPO
2. Each player turns over 2 cards and calls out the sum of the
numbers. The player with the largest sum takes all the cards.
3. In case of a tie, each tied player turns over 2 more cards and
calls out the sum of the numbers. The player with the largest
sum takes all the cards from both plays. If necessary, check
answers with a calculator.
4. The game ends when there are not enough cards left for
each player to have another turn. The player with the most
cards wins.
Lindsey turns over a red 3 and a black 6.
-3 + 6 = 3
Fred turns over a red 2 and a red 5.
-2 + (-5) = -7
3 > -7
Lindsey takes all 4 cards because 3 is greater than -7.
Variation
Each player turns over 3 cards and finds the sum.
Student Reference Book, p. 337
301_338_EMCS_S_SRB_G6_GAM_576523.indd 337
552F
Unit 6
3/15/11 10:59 AM
Number Systems and Algebra Concepts
Name
STUDY LINK
6 4a
Date
Time
Absolute Value
Plot and label the following points on the number line. The first one has
been done for you.
94
A
–10
–5
0
5
10
1.
Point A: 5.75
2.
0
Point B: _
10
3.
Point C: –6.5
4.
84
Point D: _
9
5.
Point E: –2.75
6.
1
Point F: – 8_
2
9.
| –6.5 | =
12.
1| =
| –8_
2
Find the absolute value of each of the numbers you plotted.
7.
10.
| 5.75 | =
84 | =
|_
9
8.
11.
0|=
|_
10
| –2.75 | =
y
Plot the following points on the coordinate grid.
10
13.
9
Point A: (– 3,– 6)
8
7
14.
6
Point B: (1,– 6)
5
4
15.
3
Point C: (– 3,8)
2
1
16.
Point D: (– 8,8)
10
9
8
7
6
5
4
3
2
1 0
1
1
2
3
4
5
6
7
8
9
10
2
Copyright © Wright Group/McGraw-Hill
3
Use absolute value to find the distance between
the given points. Show how you solved each
problem.
4
5
6
7
8
9
10
17.
A and B
18.
A and C
19.
C and D
192A
Name
LESSON
6 4a
Date
Time
Absolute Value Riddles
Use your knowledge of absolute value to answer the riddles.
1.
I am thinking of a number.
The absolute value of the number is 19.
The number is not equal to its absolute value.
What number am I thinking of?
2.
I am thinking of a number.
I do not have to take an opposite to find the absolute value of the number.
The absolute value of the number is 451.
What number am I thinking of?
3.
I am thinking of a number.
The absolute value of the number is not positive.
What number am I thinking of?
4.
The balance of Amber’s bank account at the beginning of the day was $150.
At the end of the day, the absolute value of the change to her account
balance was $70.
Amber deposited money today.
What is Amber’s new account balance?
The absolute value of the change in what Hal’s car is worth since he bought
it is $7,500.
Hal has not had any work done to increase what the car is worth.
Hal’s car was worth $16,000 when he bought it.
What is Hal’s car currently worth?
6.
Write and solve your own absolute value riddle below.
192B
Copyright © Wright Group/McGraw-Hill
5.